The kibana default logo exists at ./src/ui/public/images/kibana.svg under the kibana folder. As you can see, its an SVG to begin with and you can easily replace this with another custom SVG. The only caveat I would say is that whether you can do so w.r.t. copyright.

I am not sure whether you can have a kibana instance running publicly (or otherwise) that is marked as something other than kibana ... Please confirm this point before you do so